If \( y\ =\ (1+x)(1+x^2)(1+x^4) \) , then \( \displaystyle\frac{dy}{dx} \) at \( x=-1 \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 4
By the product rule, with \( (1+x^2)(1+x^4) \) as one factor:
\( \displaystyle \frac{dy}{dx}=(1+x^2)(1+x^4)+(1+x)\frac{d}{dx}\left[(1+x^2)(1+x^4)\right] \)
At x = −1, the second term vanishes because 1 + x = 0:
\( \displaystyle \left.\frac{dy}{dx}\right|_{x=-1}=(1+1)(1+1)+0=4 \)